Effects of numerical implementations of the impenetrability condition on non-linear Stokes flow: applications to ice dynamics

TL;DR

This paper compares numerical implementations of the impenetrability condition in non-linear Stokes flow models for ice dynamics, finding that the strong method is most accurate and stable for realistic glacier simulations.

Contribution

It evaluates and compares three numerical methods for enforcing the impenetrability condition in ice flow models, highlighting the advantages of the strong method.

Findings

The strong method enforces impenetrability more accurately in realistic scenarios.

The approximative method shows deviations of over 5% in 3D flow.

All methods exhibit similar convergence in manufactured solutions.

Abstract

The basal sliding of glaciers and ice sheets can constitute a large part of the total observed ice velocity, in particular in dynamically active areas. It is therefore important to accurately represent this process in numerical models. The condition that the sliding velocity should be tangential to the bed is realized by imposing an impenetrability condition at the base. We study the, in glaciological literature used, numerical implementations of the impenetrability condition for non-linear Stokes flow with Navier's slip on the boundary. Using the finite element method, we enforce impenetrability by: a local rotation of the coordinate system (strong method), a Lagrange multiplier method enforcing zero average flow across each facet (weak method) and an approximative method that uses the pressure variable as a Lagrange multiplier for both incompressibility and impenetrability. An analysis of the latter shows that it relaxes the incompressibility constraint, but enforces impenetrability approximately if the pressure is close to the normal component of the stress at the bed. Comparing the methods numerically using a method of manufactured solutions unexpectedly leads to similar convergence results. However, we find that, for more realistic cases, in areas of high sliding or varying topography the velocity field simulated by the approximative method differs from that of the other methods by $\sim 1\%$ (two-dimensional flow) and $> 5\%$ when compared to the strong method (three-dimensional flow). In this study the strong method, which is the most commonly used in numerical ice sheet models, emerges as the preferred method due to its stable properties (compared to the weak method in three dimensions) and ability to well enforce the impenetrability condition.